Here is the full list of all 17 RHYUP achievements.
Beat campaign on Easy mode
Beat campaign on Normal mode
Beat campaign on Classic mode
Get a full combo in Standard or Classic mode
Destroy all bosses in Standard or Classic campaign
Beat a previous high score
Get a perfect score on 10 songs
Get a perfect score on 10 Standard mode songs
Get a perfect score on 10 Classic mode songs
Defeat 1st Temponian Leader in Standard or Classic mode
Destroy 1st Temponian Leader in Standard or Classic mode
Defeat 2nd Temponian Leader in Standard or Classic mode
Destroy 2nd Temponian Leader in Standard or Classic mode
Defeat the Temponian Twin Leaders in Standard or Classic mode
Destroy the Temponian Twin Leaders in Standard or Classic mode
Defeat the Temponian Leader in Standard or Classic mode
Destroy the Temponian Leader in Standard or Classic mode